 display the multiplication table of a Lie algebra or a general non-commutative algebra - Maple Programming Help

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LieAlgebras[MultiplicationTable] - display the multiplication table of a Lie algebra or a general non-commutative algebra

Calling Sequences

MultiplicationTable(LieAlgebraName, keyword)

Parameters

LieAlgebraName  - (optional) name or string, the name assigned to a Lie algebra

keyword         - keyword string, one of "LieBracket", "ExteriorDerivative", "LieDerivative", "AlgebraTable"

Description

 • MultiplicationTable(LieAlgebraName, keyword) displays the form of structure equations for the Lie algebra or algebra dictated by the keyword.
 • If the keyword is "LieBracket", then the Lie brackets of the basis elements are displayed in a two-dimensional array.
 • If the keyword is "AlgebraTable", then the non-commutative products of the basis elements are displayed in a two-dimensional array.
 • If the keyword is "ExteriorDerivative", then the exterior derivatives $d\left({\mathrm{θ}}^{i}\right)$of the dual basis elements are printed.
 • If the keyword is "LieDerivative", then the Lie derivatives ${\mathrm{ℒ}}_{{e}_{i}}{\mathrm{θ}}^{j}$ of the dual 1-forms  with respect to the basis vectors are displayed in a two-dimensional array.
 • If LieAlgebraName is omitted, then the appropriate multiplication table of the current algebra is displayed.
 • The command MultiplicationTable is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form MultiplicationTable(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MultiplicationTable(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

First we initialize a 5 dimensional Lie algebra.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[5\right]\right],\left[\left[\left[2,3,1\right],1\right],\left[\left[2,5,3\right],1\right],\left[\left[4,5,4\right],1\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

Display the Lie bracket multiplication table.

 Alg1 > $\mathrm{MultiplicationTable}\left("LieBracket"\right)$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}\right]$ (2.1)

Display the exterior derivatives of the dual 1-forms.

 Alg1 > $\mathrm{MultiplicationTable}\left("ExteriorDerivative"\right)$
 ${d}{}\left({\mathrm{θ1}}\right){=}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}$
 ${d}{}\left({\mathrm{θ2}}\right){=}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}$
 ${d}{}\left({\mathrm{θ3}}\right){=}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ5}}$
 ${d}{}\left({\mathrm{θ4}}\right){=}{-}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ5}}$
 ${d}{}\left({\mathrm{θ5}}\right){=}{0}{}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}$ (2.2)

Display the Lie derivatives of the dual 1-forms.

 Alg1 > $\mathrm{MultiplicationTable}\left("LieDerivative"\right)$
 $\left[\begin{array}{ccccccc}{}& {|}& {\mathrm{θ1}}& {\mathrm{θ2}}& {\mathrm{θ3}}& {\mathrm{θ4}}& {\mathrm{θ5}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {|}& {0}& {0}& {0}& {0}& {0}\\ {\mathrm{e2}}& {|}& {-}{\mathrm{θ3}}& {0}& {-}{\mathrm{θ5}}& {0}& {0}\\ {\mathrm{e3}}& {|}& {\mathrm{θ2}}& {0}& {0}& {0}& {0}\\ {\mathrm{e4}}& {|}& {0}& {0}& {0}& {-}{\mathrm{θ5}}& {0}\\ {\mathrm{e5}}& {|}& {0}& {0}& {\mathrm{θ2}}& {\mathrm{θ4}}& {0}\end{array}\right]$ (2.3)

Example 2.

We initialize a 4 dimensional Lie algebra. Instead of using the standard default labels for the basis vectors we use and for the dual 1-forms we use .

 Alg1 > $\mathrm{L2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[4\right]\right],\left[\left[\left[2,3,1\right],1\right],\left[\left[2,4,3\right],1\right],\left[\left[4,2,4\right],1\right]\right]\right]\right):$
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2},\left[X,Y,U,V\right],\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\sigma },\mathrm{\tau }\right]\right):$

Display the Lie bracket multiplication table.

 Alg1 > $\mathrm{MultiplicationTable}\left("LieBracket"\right)$
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.4)

Display the exterior derivatives of the dual 1-forms.

 Alg1 > $\mathrm{MultiplicationTable}\left("ExteriorDerivative"\right)$
 ${d}{}\left({\mathrm{α}}\right){=}{-}{\mathrm{β}}{}{\bigwedge }{}{\mathrm{σ}}$
 ${d}{}\left({\mathrm{β}}\right){=}{0}{}{\mathrm{α}}{}{\bigwedge }{}{\mathrm{β}}$
 ${d}{}\left({\mathrm{σ}}\right){=}{-}{\mathrm{β}}{}{\bigwedge }{}{\mathrm{τ}}$
 ${d}{}\left({\mathrm{τ}}\right){=}{-}{\mathrm{β}}{}{\bigwedge }{}{\mathrm{τ}}$ (2.5)

Display the Lie derivatives of the dual 1-forms.

 Alg1 > $\mathrm{MultiplicationTable}\left("LieDerivative"\right)$
 $\left[\begin{array}{cccccc}{}& {|}& {\mathrm{α}}& {\mathrm{β}}& {\mathrm{σ}}& {\mathrm{τ}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {X}& {|}& {0}& {0}& {0}& {0}\\ {Y}& {|}& {-}{\mathrm{σ}}& {0}& {-}{\mathrm{τ}}& {\mathrm{τ}}\\ {U}& {|}& {\mathrm{β}}& {0}& {0}& {0}\\ {V}& {|}& {0}& {0}& {\mathrm{β}}& {-}{\mathrm{β}}\end{array}\right]$ (2.6)

Example 3.

We initialize the quaternions $\mathrm{ℍ}$ and display the multiplication table.

 Alg1 > $\mathrm{L3}≔\mathrm{AlgebraLibraryData}\left("Quaternions",H\right)$
 ${\mathrm{L3}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}\right]$ (2.7)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L3},\left[e,i,j,k\right],\left[\mathrm{\theta }\right]\right)$
 ${\mathrm{algebra name: H}}$ (2.8)
 Alg1 > $\mathrm{MultiplicationTable}\left("AlgebraTable"\right)$
 $\left[\begin{array}{cccccc}{}& {|}& {e}& {i}& {j}& {k}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {e}& {|}& {e}& {i}& {j}& {k}\\ {i}& {|}& {i}& {-}{e}& {k}& {-}{j}\\ {j}& {|}& {j}& {-}{k}& {-}{e}& {i}\\ {k}& {|}& {k}& {j}& {-}{i}& {-}{e}\end{array}\right]$ (2.9)